Raising AK in big blind vs 3 limpers

[elindauer]

In this post, I'll be discussing, again, the old question of whether or not to raise AK from the blinds. Since this is a very complicated topic, I'm going to try to limit the discussion in the following ways:

1. You are the big blind facing 3 limpers. the small blind folds.

2. All limpers have exactly the following hand range:
any ace. any suited connector. any pair below TT. any two cards totalling 20 or more.

3. the framework for discussion: we will calculate 4 numbers
- EV of AK in unraised pot when it improves
- EV of AK in raised pot when it improves
- EV of AK in unraised pot when it misses
- EV of AK in raised pot when it misses

4. the EV of our hand when we hit will be determined by calculating the EV of the draws against us, subtracting this from the pot. See this post for an explanation of why I think this is best.


So, for the purposes of this thread, statements like "AK has an equity edge preflop" and "AK makes more money on every call" are simply inadmissable as they are in violation of rule 4. If this is the way you prefer to think about profit, I respectfully request that you choose not to post in this thread.



Now, I'll get us started with some rough calculations of the 4 key numbers. Please reply with suggested improvements for any of these calculations.

- EV of AK in unraised pot when it improves
It's not uncommon to find draws out against AK that will lose money in this situation. I'll say that players calling down with weaker aces have -EV draws and balance out the EV paid out to hands like flush draws. Factor in the occassional flopped set and maybe we have 90% equity in the pot.

EV = 4.95 SB

- EV of AK in raised pot when it improves

In this spot, the draws out against us have all gotten more valuable, so we can't possibly give the whole pot to AK like we did above. Implied odds have gone up some as there is no way we will almost never get away from this hand. More draws can call profitably. I'll take a stab and say the draws suck out 70% of the 8.5 SB pot on average.

EV: 5.95 SB

- EV of AK in unraised pot when it misses

tough one. AK often can't call a bet, but he sometimes finds ways to show his hand down unimproved to win. I'll call it 10% pot equity in this 5.5 SB pot for it's draw.

EV = .55 SB

- EV of AK in raised pot when it misses

The bigger pot makes the draw more valuable. He'll have profitable opportunities to show down unimproved more often, not to mention more opportunities to draw to the overcards. Maybe 15% equity for the draw.

EV: .825 SB



I have made a good faith effort to stab at these numbers. In fact, I don't even know, as I type this, whether they suggest a raise or a check is correct preflop. I expect all of these numbers to become more clearly defined during this thread, but for starters, here are the results:

EV (check) = 1/3 (.55) + 2/3 (4.95) - .5 ~= 3 SB
EV (raise) = 1/3 (.825) + 2/3 (5.95) - 1 ~= 3.25 SB

These numbers are pretty close, clearly the margin of error in my guesses is greater.

Thanks.
Eric


PS. If anyone can calculate the odds of AK improving given the hand ranges described, that would be helpful.

[SeaEagle]

It looks like you're using 5.5 for 3 of the calculations and 8.5 for the other. You probably want to double check this. It also looks like you're subtracting investments/bets of .5 and 1 sbs, how did you figure this?

Those issues aside, I'd ask you to clarify a couple of things:

One, you are estimating that AK wins 20% less when the pot is bigger?

Two, in the other thread you calculated from the final pot (and that was very important to the outcome). It looks to me like here you are not putting in the extra bets that happen after the flop, and simply calculating from the pot at the beginning of the flop. Am I missing something?

[elindauer]

[ QUOTE ]
It looks like you're using 5.5 for 3 of the calculations and 8.5 for the other. You probably want to double check this.

[/ QUOTE ]

I don't see it. Which calculation is off?

[ QUOTE ]
It also looks like you're subtracting investments/bets of .5 and 1 sbs, how did you figure this?

[/ QUOTE ]

Mistake. For some reason I often mix small and big bets in my posts. That would make the final EVs:

check = 2.5 SB
raise = 2.25 SB

or actually favoring a check.


[ QUOTE ]
One, you are estimating that AK wins 20% less when the pot is bigger?

[/ QUOTE ]

Yes, AK improved wins less in the big pot than in the small pot. I would defend this concept, although perhaps not the absolute value, by pointing out that in the big pot, more players will draw, cutting down the win rate. Also, the weak draws that come along are doing so correctly, so instead of adding to your win, they are instead taking some of the pot from you.

[ QUOTE ]
Two... you are not putting in the extra bets that happen after the flop, and simply calculating from the pot at the beginning of the flop. Am I missing something?

[/ QUOTE ]

Yes. I'm factoring future bets into the implied odds / reverse implied odds of the various draws I'm guestimating will be out against you. I concede that so far I haven't been rigorous about estimating the probability of the various draws or their EV, opting instead to just throw out estimates and hope for some help.

-Eric

[Jake (The Snake)]

[ QUOTE ]

I don't see it. Which calculation is off?

[/ QUOTE ]

The pot size for unraised should be 4.5.

3 limpers + SB + BB

[ QUOTE ]
Yes. I'm factoring future bets into the implied odds / reverse implied odds of the various draws I'm guestimating will be out against you. I concede that so far I haven't been rigorous about estimating the probability of the various draws or their EV, opting instead to just throw out estimates and hope for some help.

[/ QUOTE ]

I don't see how you are doing this either. You are taking the % you think we will end up winning and multiplying only by the preflop pot size as far as I can tell.

[elindauer]

[ QUOTE ]
The pot size for unraised should be 4.5.

[/ QUOTE ]

Ah, thanks.

[ QUOTE ]
[ QUOTE ]
Yes. I'm factoring future bets into the implied odds / reverse implied odds of the various draws I'm guestimating will be out against you. I concede that so far I haven't been rigorous about estimating the probability of the various draws or their EV, opting instead to just throw out estimates and hope for some help.

[/ QUOTE ]

I don't see how you are doing this either. You are taking the % you think we will end up winning and multiplying only by the preflop pot size as far as I can tell.

[/ QUOTE ]

It's not clear because I'm just hand waving at the implied odds and calling it a day, but the key is to understand that the percentage I am giving is not an estimate of the probability the AK will be good, but an estimate of the percentage of the pot this hand will capture.

See the thread in the original post of the method of calculation for more explanation of this. In that thread, AK wins 91% of the time, but captures 98% of the pot. To the extent that just throwing out numbers can be considered doing any math at all, that's exactly what I'm doing here.

Good luck.
Eric

[SeaEagle]

[ QUOTE ]
I concede that so far I haven't been rigorous about estimating the probability of the various draws or their EV, opting instead to just throw out estimates and hope for some help.


[/ QUOTE ]
Here's some PS numbers for the hand ranges you give:
PF equity: AK - 31%. Others 23% each.
Ac8d7d equity: AK - 53%. Others 16% each
Ac7d2h equity: AK - 61%. Others 13% each
QcJc7c equity: AK - 29%. Others 24% each
9s7c5d equity: AK - 21%. Others 26% each

I don't think this tells us much, but it gives us some starting points. AK will be win more than 53-61% when he hits because opponents will fold. AK will win less than 29-21% when he misses because he will fold.

[elindauer]

Thanks SeaEagle, this is very helpful. Some other things that would make this even better:

- when we miss, how often is AK the best hand?
- when we hit, how common are boards similar to the ones you listed?
- when we hit, what kinds of draws are typically against on each of these boards?

I'll try to incorporate these suggestions along with the math errors you pointed out into a new post and put some new best guess estimates of the values of checking and raising.

Thanks,
Eric

[SeaEagle]

[ QUOTE ]
when we miss, how often is AK the best hand?

[/ QUOTE ]
Unfortunately, I don't how to narrow this down very well. AK will only win a showdown about 25% of the time but about 24% of the time he'll improve between now and the river. SO that doesn't tell me much.

On the other hand, if I look at from the other side and try to figure out the odds that 3 random hands miss the flop, assuming each hand misses the flop 2/3 of the time all 3 will miss the flop about 30% of the time, so that doesn't help either.

[ QUOTE ]
- when we hit, how common are boards similar to the ones you listed?


[/ QUOTE ]
I put in a bunch of different boards and took the ones that showed the highest and lowest equity sitations for AK, so these are intended to show a plausible range of draw equities.

[ QUOTE ]
when we hit, what kinds of draws are typically against on each of these boards?


[/ QUOTE ] This would be really valuable, but I have no way of even guessing at this one. For instance, the Ac8d7d flop is all over the place with flush, straight, Ax 3-out draws, and 8x/7x 5-out draws.

[chief444]

Eric,

I've made this comment about 10 times already but we were discussing against VERY loose opponents. These opponents at least call the flop with any piece whether it's correct or not. So you will "gain" more postflop in a smaller pot against these weak draws that call incorrectly but you don't "lose" (so to speak) any in the bigger pot against these opponents since they would have called anyway. This is sort of important to this analysis as there is a better case for checking against better opponents. It's also important relative to your calculations because you're EV doesn't change...at least not nearly as much as your estimating. Not against very loose opponents. People who limp with any ace don't fold the flop unless they have about no equity anyway.

I commend your efforts. I sincerely do.

Matt

[SeaEagle]

[ QUOTE ]
I've made this comment about 10 times already but we were discussing against VERY loose opponents.

[/ QUOTE ]
Chief, with all due respect, Eric has worked really hard on this so far. I'd really like to at least get a working model that we can generally agree comes close and then see if we can modify parameters/values for factors such as opponent looseness and what not. Sound like a plan?